Systematics of quarkonium production at the LHC and double parton fragmentation

In this paper, we discuss the systematics of quarkonium production at the LHC. In particular, we focus on the necessity to sum logs of the form $\log (Q/p_{\perp })$ and $\log (p_{\perp }/m_Q)$. We show that the former contributions are power suppressed, while the latter, whose contribution in fragmentation is well known, also arise in the short distance (i.e., nonfragmentation) production mechanisms. Though these contributions are suppressed by powers of $m_Q/p_{\perp }$, they can be enhanced by inverse powers of $v$, the relative velocity between heavy quarks in the quarkonium. In the limit $p_{\perp }\gg m_Q$, short-distance production can be thought of as the fragmentation of a pair of partons (i.e., the heavy quark and antiquark) into the final state quarkonium. We derive an all-order factorization theorem for this process in terms of double parton fragmentation functions and calculate the one-loop anomalous dimension matrix for the double parton fragmentation functions.

Figure 1

(a) The short distance production mechanism with the quark pair produced in a color-singlet state. (b) Gluon fragmentation production contribution to hadroproduction. The two soft gluons emitted via E1 transitions are suppressed by $v^4$.

Figure 2

A typical contribution to gluon fragmentation with two hard central jets. The lines going across the cut are integrated out, while the uncut lines represent the fragmenting gluon.

Figure 3

Matching $O_{qq}^G$ at leading order. On the left is the square of the full theory amplitudes, and on the right is the matrix element of $O_{qq}^G$. At this order the matching coefficient is proportional to ${\alpha }_s^2(Q)$.

Figure 4

Matching of $O_{qq}^{QQ}$ onto the full theory at leading order. On the left are two leading-order Feynman diagrams that contribute to the production of a $Q\overline{Q}$ pair from an incoming $q\overline{q}$ pair. On the right is the tree-level matrix element of $O_{qq}^{QQ}$. The dashed lines are incoming and outgoing collinear light quark lines, and the dashed double lines are incoming and outgoing heavy quarks. At this order the matching coefficient is proportional to ${\alpha }_s^3(Q)$.

Figure 5

The diagrams we need for the one-loop running. Not shown are the diagrams that are mirror images with respect to the horizontal and vertical axes. Diagram A reflected about a horizontal (vertical) is denoted in the text by $\widehat{A}(\overline{A})$.